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Introduction to MATLABUE Numerische Mathematik 1

Wintersemester 2015

Scott Congreve

Fakultät für Mathematik

Universität Wien

Oskar-Morgenstern-Platz 1

A-1090 Wien

Contents

1 Introduction 11.1 Overview of the UI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basics of the Command Window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Documentation & Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Basic Mathematics 32.1 Scalar Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Number Format & Special Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Vectors & Matrices 63.1 Defining Matrices & Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Vector & Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3.1 Matrix/Vector Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3.2 Basic Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3.3 Element-wise Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.4 Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3.5 Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.1 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.2 Matrix Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Strings 164.1 Formatting Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Displaying Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Graphics 175.1 Plot Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 Annotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.3 3D Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6 Programming 226.1 Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 Logical Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.4 Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.4.1 for Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4.2 while Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.5 if-else Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.6 switch Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.7 Function Handles & Anonymous Functions . . . . . . . . . . . . . . . . . . . . . . . 31

7 Structures 32

8 Error Handling 338.1 Understanding Error Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Generating Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.3 Debugging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Literature & Resources 35

List of Code Examples

1 help Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Evaluating (22)3 and 22

3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Demonstration of requirement for multiplication symbol . . . . . . . . . . . . . . . 34 Short format for real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Long format example; also demonstrates rounding error . . . . . . . . . . . . . . . 46 MATLAB constants/special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Definition and usage of the variables x and y . . . . . . . . . . . . . . . . . . . . . . 58 Clearing only the x and y variables from the workspace . . . . . . . . . . . . . . . . 59 Definition of complex number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 Example of calling a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 Example of calling a function with multiple arguments . . . . . . . . . . . . . . . . 612 Generating matrices/vectors directly . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 Matrix/vector concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 Generating a vector containing a sequence . . . . . . . . . . . . . . . . . . . . . . . 715 Example usage of linspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816 Example usage of matrix construction functions . . . . . . . . . . . . . . . . . . . . 817 Example usage of diag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818 Generating a complex matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919 Using reshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 920 Basic matrix/vector indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1021 Vector indices for matrix/vector indexing . . . . . . . . . . . . . . . . . . . . . . . . 1022 Extracting a complete row or column from a matrix . . . . . . . . . . . . . . . . . . 1023 Changing values in a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124 Deleting vector and matrix entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Obtaining matrix/vector size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226 Basic arithmetic with matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228 Element-wise arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 Matrix transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1430 Solving linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431 Handling multiple function returns (calculating eigenvalues/eigenvectors) . . . . 1632 String example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633 String concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634 Formatting numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1735 Displaying text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1736 Basic plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1837 Annotating a plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1938 3D line (parametric) plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939 3D surface plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2040 3D mesh/contour plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2141 Sample script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2342 Sample script output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2343 Default function structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2444 Simple function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2445 Calling the simple function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2546 Simple function changed for vector/matrix inputs . . . . . . . . . . . . . . . . . . . 2547 Calling functions with vector arguments . . . . . . . . . . . . . . . . . . . . . . . . . 2548 Sub-function example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2549 Example of rounding error in comparisons . . . . . . . . . . . . . . . . . . . . . . . 2650 Logical operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2751 Using logical indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

52 find indices of all values < 0.02 in matrix . . . . . . . . . . . . . . . . . . . . . . . . 2853 for loop structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2854 Factorial calculation with for loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2855 Result of factorial calculation with for loop . . . . . . . . . . . . . . . . . . . . . . . 2856 while loop structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2957 while loop example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2958 Result of while loop example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2959 if-elseif-else structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2960 if example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3061 switch structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3062 switch example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3063 Taking and using function handle of sin . . . . . . . . . . . . . . . . . . . . . . . . . 3164 Calling ezplot with function handle of sin . . . . . . . . . . . . . . . . . . . . . . . . 3165 Calling ezsurf with handle to own function . . . . . . . . . . . . . . . . . . . . . . . 3166 3D mesh/contour plot using anonymous functions . . . . . . . . . . . . . . . . . . 3167 Generating structure using struct function . . . . . . . . . . . . . . . . . . . . . . . 3268 Generating and reading structure directly . . . . . . . . . . . . . . . . . . . . . . . . 3269 Accessing structure array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3270 Example error messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3371 Function with error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3372 Running function with error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3373 Generating errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3474 Executing a function with error checking . . . . . . . . . . . . . . . . . . . . . . . . 34

List of Figures

1 Overview of the User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Basic plotting result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Annotated plotting result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 3D line (parametric) plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 3D surface plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 Result of view(2) on 3D surface plot . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 3D mesh/contour plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Debug tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

INTRODUCTION TO MATLAB 1

1 Introduction

MATLAB is a high-level programming language and interactive environment designed for nu-merical computation. It has both an interactive console for executing individual commands, andsupport for writing full scripts (programs).

In this section, we shall give a brief overview of the MATLAB program. In Sections 2–5 weshall cover various basic MATLAB commands, using only the interactive console. Section 6 willcover using MATLAB to write scripts and functions. Finally, Sections 7 & 8 will cover a couple ofmore advanced topics.

1.1 Overview of the UI

Command

Window

Command

History

Current

Folder

Menu Bar Layout Workspace

Figure 1: Overview of the User Interface

Upon launching MATLAB the main window is displayed. The main area of this window isusually the Command Window, this is an interactive console where MATLAB commands can beentered at the prompt and results seen immediately. The default MATLAB window also displaysa Workspace panel, which lists all the current variables (see Section 2.3), and a Current Folder panellisting all files in the directory MATLAB considers “current” (see Section 1.2). Another usefulpanel, which is often not shown by default, is the Command History panel, which lists all recentcommands entered into the Command Window. This can be shown from the Layout button on theHome tab of the menu bar.

The MATLAB window is completely customisable. Panels can be shown or hidden from theLayout button on the Home tab of the menu bar. Panels can also be dragged to different positions,placed into tabs with each other and even undocked into separate windows. Each panel has a

2 UE NUMERISCHE MATHEMATIK 1

small downward pointing arrow in the top right corner, which opens a menu containing variouscustomisation options.

1.2 Basics of the Command Window

The Command Window consists of a prompt (>>) at which MATLAB commands can be entered.Results from each MATLAB command ran is also displayed in the Command Window. You canclear the current command window of all output by entering the clc command into the CommandWindow.

MATLAB keeps a history of all commands entered (these can be seen and selected to run againfrom the Command History panel). You can also use the and arrow keys on the keyboardto scroll backwards or forwards, respectively, through the history of commands entered. If youstart to type a command and then press the or arrow keys then MATLAB will only scrollthrough commands which start with the text already entered.

When you try to execute a MATLAB function it searches in a list of paths for the a file con-taining the definition of that function. By default this list consists of a set of built-in MATLABdirectories and also the current directory according to MATLAB. Usually when you start MATLABthis will be the MATLAB subdirectory in your HOME or Documents folder. When we come towrite scripts and functions (see Section 6) your current directory will need to be the same as thedirectory where you save these files in order to be able to run them. Entering the command, cd,on its own lists the current directory. You can also change the current directory by using

>> cd path

where path is the path to change to. Directories in a path are separated by a / and a special ..directory can be used to change to the parent directory. For example, if my current directory isUsers congreve Documents MATLAB then calling

>> cd ../TestFolder

will change the current directory to Users congreve Documents TestFolder. Note that ifany folder contains a space you should surround the path with single quotation marks:

>> cd ’../Folder With Space/Folder’

You can see a list of all files in the current directory with the ls command, or only MATLABspecific files with the what command.

To exit MATLAB type exit at the prompt.

1.3 Documentation & Help

MATLAB has built-in documentation for all its commands, functions and syntax. This documen-tation can be viewed in two different ways. The first way is a graphical help window which canbe launched by selecting the Help Documentation menu, or by entering the command doc into theCommand Window. The doc command can be followed by a name (usually of a MATLAB function),in which case the documentation window is automatically launched to view the documentationfor that command. For example,

>> doc sin

will open the documentation for the sin function.The second method is a text-based help displayed directly in the Command Window. To view

the contents for this help type help into the Command Window. Again you may add a name of afunction, command, toolbox, etc. to display the help for that command.


>> help sinsin Sine of argument in radians.

sin(X) is the sine of the elements of X.

See also asin, sind.

Other functions named sin

Reference page in Help browserdoc sin

Code Example 1: help Example

2 Basic Mathematics

MATLAB is designed to perform mathematical operations on scalars, vector and matrices. Weshall start by looking at the basic scalar mathematics.

2.1 Scalar Arithmetic

In MATLAB you can enter mathematical statements to solve in an ALMOST identical way to howyou write them on paper. MATLAB supports five basic scalar mathematical operators. These are+ (for addition), - (for subtraction), * (for multiplication), / (for division), and ^ (for raising to apower). There is also a left division operator \ which divides the second term by the first. Usingthese basic commands you can use MATLAB as a calculator; i.e., entering

>> 5^2+9.5-11*2ans =

12.5000

displays the result of 52 + 9.5− 11× 2. MATLAB follows the basic mathematical rules for prece-dence; ^ is evaluated first, then * and /, and then + and -. Operators of the same precedenceare evaluated with left-to-right associativity — the first operator from the left is evaluated first.Brackets ( ) can be used to specify order of evaluation.

>> 2^2^3ans =

64

>> 2^(2^3)ans =

256

Code Example 2: Evaluating (22)3 and 223

Note that unlike in normal mathematics the multiplication symbol must be used wherever mul-tiplication is required.

>> 2(4+5)2(4+5)|

Error: Unbalanced or unexpected parenthesis or bracket.

>> 2*(4 + 5)ans =

18

Code Example 3: Demonstration of requirement for multiplication symbol


2.2 Number Format & Special Constants

By default in MATLAB all numbers generated are of double type. The technically of what thismeans exactly is beyond the scope of this course, but essentially each number is stored with thecomputer’s memory as a 64-bit binary floating-point number. This means that it can representfloating-point numbers, but we do have to allow for small rounding errors in computations asnot all decimal floating point numbers can be accurately represented within the number of bits(the only real important point about this we shall discuss in Section 6.3).

MATLAB, like most programming languages, allows us to enter numbers in an exponential(base 10) form. Essentially entering 1.5e-10 is short-hand for 1.5×10−10, and 7.95e5 is short-handfor 7.95× 105.

When MATLAB displays a floating point number it usually displays it in short form (fourdecimal places) in either normal or exponential form.

>> 190.2ans =190.2000

>> 1909.205ans =

1.9092e+03

Code Example 4: Short format for real numbers

Notice that in the last case we lost some of the number in the display (it is still there but MAT-LAB has not displayed the result). We can ask MATLAB to display all results in long form (15decimal places) with the command format long, and we can switch back to short format withformat short. More formats exist as well, use help format to see the complete list. You can setthe default format in the Preferences window in MATLAB (under MATLAB Command Window ).

>> format long>> 19.2ans =19.199999999999999

>> 1909.205ans =

1.909205000000000e+03

Code Example 5: Long format example; also demonstrates rounding error

MATLAB has a number of built-in constants and special values that can be used (and dis-played). pi returns the constant value for π while eps returns the difference between 1 and nextlargest double-precision floating-point number. Double-precision numbers have three specialnumbers, inf/Inf and -inf/-Inf represent∞ and −∞, respectively, while nan/NaN represents aspecial Not a Number value.

>> 0/0ans =

NaN>> 1/0ans =

Inf>> -1/0ans =-Inf

>> pians =

3.141592653589793>> epsans =

2.220446049250313e-16

Code Example 6: MATLAB constants/special values


2.3 Variables

As a programming language MATLAB has a concept of variables that be used to store values.Assigning a value to a variable is done via the assignment = operator. When assigning a variable,the value stored into the variable is output in the Command Window; this can be suppressed byending the command with a semicolon (;). Variables can be used in expressions similar to basicmathematics and entering a variable name on its own at the prompt (without a trailing semicolonat the end) will output the variables value.

>> x = 2^2x =

4

>> 5*x+9ans =

29

>> y = 9;>> yy =

9

Code Example 7: Definition and usage of the variables x and y

Note again that we need to explicitly use the multiplication operator for calculating 5x+ 9.Variables names in MATLAB must start with a letter and can contain only letters, numbers

and the underscore (_) character. Note that by letter we mean the basic 26 English letters (so noaccented letters or the “scharfes S”). Ideally, variable names should be self-explanatory wherepossible and also note that all variable names are case sensitive; i.e., A and a are different variables.Note that defining a variable with the same name as a built-in MATLAB constant (pi, eps, etc.) orfunctions will hide the definition of that function or constant, so this should be avoided. There isalso a special variable called ans, which stores the result of the last command entered if the resultis not saved to a variable.

When working in the Command Window all variables defined are saved in the Workspace. Youcan see the list of all variables in the Workspace panel or by entering the who or whos command. Youcan clear all variables from the current workspace with the clear command; alternatively, you canclear a single or list of variables by enter the names of the variables after the clear command.

>> clear x y

Code Example 8: Clearing only the x and y variables from the workspace

2.4 Complex Numbers

MATLAB supports complex numbers as well as real numbers. To specify an imaginary numberyou use i or j either directly or as a suffix to a number. For example, to generate the complexnumber z = 5 + 4i:

>> z = 5+4iz =

5.0000 + 4.0000i

Code Example 9: Definition of complex number

When used in scripts (see Section 6) newer versions of MATLAB will produce warnings aboutusing i or j without a number prefix and will advise the use of 1i and 1j instead.


2.5 Functions

MATLAB has a large collection of built-in functions for mathematical operations. Functions arecalled by giving the name of the function followed by the arguments within brackets after thename. For example, to calculate sin π/2 we enter the command:

>> sin(pi/2)ans =

1

Code Example 10: Example of calling a function

Some functions can take more than one argument; in this case, we enter the arguments separatedby a comma.

>> min(pi, 3)ans =

3

Code Example 11: Example of calling a function with multiple arguments

You can store the results into a variable as normal. Some functions are able to return more thanone result, which we shall see in action in Section 3.4.2.

Below is a non-exhaustive list of basic mathematical functions (for complex and/or real num-bers). Enter help elfun for a more complete list.

sin, cos, tan, cot, sec, csc Trigonometric functionsasin, acos, atan, acot, sec, csc Inverse trigonometric functionssinh, cosh, tanh, coth, sech, csch Hyperbolic functions

asinh, acosh, atanh, acoth, asech, acsch Inverse hyperbolic functionsabs Absolute value |x|exp Exponential function ex

log, log10, log2 Logarithmic function (base e, 10 and 2)fix, floor, ceil, round Round: to zero, down, up, nearest integer.

sqrt, nthroot Square and nth root.angle Phase angle of a complex numberconj Complex conjugate of a complex number

real, imag Real/imaginary parts of complex number

3 Vectors & Matrices

So far we have only dealt with scalar values; however, MATLAB supports matrices and vectorsas well. In this section we shall discuss the basics of the matrix support in MATLAB.

3.1 Defining Matrices & Vectors

The basic method to create a vector or matrix in MATLAB is to use square brackets [] containinga list of numbers to place in the matrix. Each row of a matrix is a list of numbers separated byeither a space and/or a comma, and each row is separated by a semi-colon ;. For example, thematrix, row vector and column vector,

A =

1 9 7−3 8 02 −7 −9

, x =(5 −8 0

), and y =

−236

,

respectively, are generated with:


>> A = [1 9 7; -3 8 0; 2 -7 -9]A =

1 9 7-3 8 02 -7 -9

>> x = [5 -8 0 9]x =

5 -8 0 9

>> y = [-2; 3; 6]y =

-236

Code Example 12: Generating matrices/vectors directly

This notation is really a concatenation of matrices/vectors. The space/comma concatenatescolumns and the semi-colon concentrates rows. It is, therefore, possible to concatenate matri-ces into larger matrices using this notation, providing the sizes are compatible.

>> B = [A y; x]B =

1 9 7 -2-3 8 0 32 -7 -9 65 -8 0 9

Code Example 13: Matrix/vector concatenation

You can also generate a row vector of a sequence of numbers using the start:step:end orstart:end syntax, where start is the first number in the sequence, step is the difference betweenelements in the sequence (in the form without step this defaults to 1), and end is the largestnumber (for a positive step) or smallest number (for a negative step) that can be contained in thesequence.

>> 1:4ans =

1 2 3 4

>> 1:0.5:3ans =

1.0000 1.5000 2.0000 2.5000 3.0000

>> 1:2:6ans =

1 3 5

>> 1:-1:6ans =

Empty matrix: 1-by-0

>> 7:-1:1ans =

7 6 5 4 3 2 1

>> 0:0.2:1ans =

0 0.2000 0.4000 0.6000 0.8000 1.0000

Code Example 14: Generating a vector containing a sequence

As can be seen the end value is not always included in the sequence. Generating a sequence ofequally distributed values, which includes both the start and end values can be done with the


linspace(start, end, no_points) function. Here, no_points is the number of items in the rowvector, including the start and end.

>> linspace(1,2,6)ans =

1.0000 1.2000 1.4000 1.6000 1.8000 2.0000

>> linspace(1,0,6)ans =

1.0000 0.8000 0.6000 0.4000 0.2000 0

Code Example 15: Example usage of linspace

MATLAB has several basic functions for generating matrices. The eye function generates anidentity matrix (ones on the diagonal, zero elsewhere), the zeros function generates a matrix ofzeros, and the ones function generates a matrix of ones. All three functions can take a single scalarargument (N ), in which case aN ×N matrix is generated, or two scalar arguments (N andM ), inwhich case a N ×M matrix is generated. These functions can also take a single vector argument(where the vector contains two values [N M ]), which also generates a N ×M matrix. The rand

and randn functions generate a matrix of uniformly or normally distributed random numbers,respectively, between 0 and 1.

>> eye(4)ans =

1 0 0 00 1 0 00 0 1 00 0 0 1

>> zeros(3)ans =

0 0 00 0 00 0 0

>> ones(4)ans =

1 1 1 11 1 1 11 1 1 11 1 1 1

>> ones(4,3)ans =

1 1 11 1 11 1 11 1 1

Code Example 16: Example usage of matrix construction functions

A diagonal matrix can be generated with the diag function. This function takes a vector andplaces the entries on the leading diagonal of a matrix. A second optional scalar integer argumentcan also be specified, which allows the vector to be placed on a different diagonal. A valueof 0 indicates the leading diagonal, a positive number indicates a diagonal above the leadingdiagonal (with 1 being the diagonal immediately above the leading diagonal) and a negativenumber indicates a diagonal below the leading diagonal.

>> diag([1 2 3 4])ans =

1 0 0 00 2 0 00 0 3 00 0 0 4


>> diag([1 2 3 4],1)ans =

0 1 0 0 00 0 2 0 00 0 0 3 00 0 0 0 40 0 0 0 0

>> diag([1 2 3 4],-1)ans =

0 0 0 0 01 0 0 0 00 2 0 0 00 0 3 0 00 0 0 4 0

Code Example 17: Example usage of diag

You can create a complex vector/matrix from two real vector/matrices (representing the realand imaginary parts) by using the complex(real,imag) function, where real and imag are realmatrices representing the real and imaginary parts.

>> complex(ones(3),eye(3))ans =

1.0000 + 1.0000i 1.0000 + 0.0000i 1.0000 + 0.0000i1.0000 + 0.0000i 1.0000 + 1.0000i 1.0000 + 0.0000i1.0000 + 0.0000i 1.0000 + 0.0000i 1.0000 + 1.0000i

Code Example 18: Generating a complex matrix

Matrices can be reshaped into a matrix of a different size using the reshape function. Thisfunction takes a matrix to reshape as the first argument followed by a matrix size (as either asingle vector of two elements or as two arguments). In the form where the new size is specifiedas two argument you can use an empty vector [] to allow MATLAB to automatically calculatethe size of that dimension. Note that the number of elements in the reshaped matrix must be thesame as in the original matrix. When reshaping a matrix MATLAB works down columns firstwhen reading the elements (and places them in the new matrix in the same way).

>> A = rand(4,2)A =

0.7363 0.44230.3947 0.01960.6834 0.33090.7040 0.4243

>> reshape(A,[3,3])Error using reshapeTo RESHAPE the number of elements must not change.

>> reshape(A,[2 4])ans =

0.7363 0.6834 0.4423 0.33090.3947 0.7040 0.0196 0.4243

>> reshape(A,3,[])Error using reshapeProduct of known dimensions, 3, not divisible into total number of elements, 8.

>> reshape(A,2,[])ans =

0.7363 0.6834 0.4423 0.33090.3947 0.7040 0.0196 0.4243

Code Example 19: Using reshape


3.2 Indexing

Each element in a vector can referred to by using an index notation starting from 1 for the firstvalue (note that this is different to some other programming languages which start from 0). In order toaccess an item of the vector you place the index to access within brackets after the variable name.Matrices require two indices, the first is the row index and the second is the column index (It ispossible to use only one index for matrices, in this case the index counts down the first column,then the second, etc.). A special value of end can be used to index the last value.

>> A = rand(4)A =

0.8147 0.6324 0.9575 0.95720.9058 0.0975 0.9649 0.48540.1270 0.2785 0.1576 0.80030.9134 0.5469 0.9706 0.1419

>> A(5)ans =

0.6324

>> A(2,3)ans =

0.9649

>> x = 1:5x =

1 2 3 4 5

>> x(3)ans =

3

>> x(end)ans =

5

Code Example 20: Basic matrix/vector indexing

For an index you can also specify a vector of indices, in which case the result is a vector con-taining just those values. This vector index can be specified as a normal index or using thestart:step:end/start:end notation. The special end value can be used in this range specifier.

>> x([1 3])ans =

1 3

>> A(2:3,2:3)ans =

0.0975 0.96490.2785 0.1576

>> A(2,2:end)ans =

0.0975 0.9649 0.4854

>> A(2,end:-1:2)ans =

0.4854 0.9649 0.0975

Code Example 21: Vector indices for matrix/vector indexing

You can also use a single colon : in an index location to indicate all values.

>> A(2,:)ans =

0.9058 0.0975 0.9649 0.4854


>> A(:,3)ans =

0.95750.96490.15760.9706

Code Example 22: Extracting a complete row or column from a matrix

You can also change individual elements in a matrix by indexing as above and using as theleft-hand side of the assignment operator =. When indexing more than one element the value youassign must either by a scalar (value is assigned to all elements) or a vector/matrix of the samesize as the indexed sub-matrix.

>> A(2,3) = 1A =

0.8147 0.6324 0.9575 0.95720.9058 0.0975 1.0000 0.48540.1270 0.2785 0.1576 0.80030.9134 0.5469 0.9706 0.1419

>> A(end,2:3) = 0.5A =

0.8147 0.6324 0.9575 0.95720.9058 0.0975 1.0000 0.48540.1270 0.2785 0.1576 0.80030.9134 0.5000 0.5000 0.1419

>> A(end,2:3) = [0.25 0.2 1]Subscripted assignment dimension mismatch.

>> A(end,2:3) = [0.25 0.7]A =

0.8147 0.6324 0.9575 0.95720.9058 0.0975 1.0000 0.48540.1270 0.2785 0.1576 0.80030.9134 0.2500 0.7000 0.1419

Code Example 23: Changing values in a matrix

You can also delete values from a vector/matrix by assigning to them the empty matrix (thebrackets [] with no contents). In the case of a matrix you can only delete complete rows orcolumns.

>> A(1,:) = []A =

0.9058 0.0975 1.0000 0.48540.1270 0.2785 0.1576 0.80030.9134 0.2500 0.7000 0.1419

>> x(2:3) = []x =

1 4 5

Code Example 24: Deleting vector and matrix entries

3.3 Vector & Matrix Operations

We shall now discuss the basic operations that be used on matrices or vectors.

3.3.1 Matrix/Vector Size

MATLAB has a built-in function size which returns a vector containing the dimension of a ma-trix. The first value in the vector is the number of rows in the matrix, and the second is the


number of columns. When run on a vector one of these values will be 1 (dependant on if thevector is a column or row vector). MATLAB also has a length function, which returns a singlevalue denoting the size of the largest dimension of the matrix. This function is most useful forvectors, as it returns the length of the vector.

>> size(A)ans =

3 4

>> size(x)ans =

1 3

>> length(A)ans =

4

>> length(x)ans =

3

Code Example 25: Obtaining matrix/vector size

Note that as size returns a two-value vector you can use the result as an argument to the matrixconstruction functions (see Section 3.1) to construct a matrix the same size as an existing matrix.

3.3.2 Basic Arithmetic

Addition + and - are defined as normal for vectors and matrices. It is important to note thatan error will occur if you try to apply these operators to matrix/vectors of different size. Youcan, however, apply these operators to a scalar and a vector/matrix. In this case the scalar isautomatically converted into a matrix/vector of the correct size filled with the scalar value

>> B = rand(size(A))B =

0.4218 0.9595 0.8491 0.75770.9157 0.6557 0.9340 0.74310.7922 0.0357 0.6787 0.3922

>> A+Bans =

1.3276 1.0570 1.8491 1.24311.0427 0.9342 1.0916 1.54341.7056 0.2857 1.3787 0.5341

>> A+1ans =

1.9058 1.0975 2.0000 1.48541.1270 1.2785 1.1576 1.80031.9134 1.2500 1.7000 1.1419

Code Example 26: Basic arithmetic with matrices

The multiplication operator * can be used to multiply every value of a matrix/vector by a scalarvalue, or to perform matrix multiplication (providing the two matrices are of compatible sizes).Note that the sequence vector notation generates row vectors, so these may need to be transposed(Section 3.3.4).

>> A*2ans =

1.8116 0.1951 2.0000 0.97080.2540 0.5570 0.3152 1.60061.8268 0.5000 1.4000 0.2838


>> A*BError using *Inner matrix dimensions must agree.

>> C = rand(4,3)C =

0.6555 0.2769 0.69480.1712 0.0462 0.31710.7060 0.0971 0.95020.0318 0.8235 0.0344

>> A*Cans =

1.3319 0.7522 1.62720.2677 0.7223 0.35391.1402 0.4493 1.3840

>> z = [1;2;3;4]z =

1234

>> A*zans =

6.04244.35794.0809

Code Example 27: Matrix multiplication

The division / and left division \ operators can be used to divide each element of a matrix by ascalar (see Section 3.3.5 for another use of these operators). The power operator ^ can be usedto raise a matrix by a scalar value (we shall only worry about positive integers here — in whichcase this operator is equivalent to repeated matrix multiplication of the matrix with itself)

3.3.3 Element-wise Arithmetic

MATLAB also supports element-wise arithmetic operators. When applied to two matrices/vec-tors of the same size they apply the matching scalar operation to the elements with the sameindex. If only one of the arguments of the element-wise operator is a scalar then the scalar isautomatically converted into a matrix/vector of the correct size filled with the scalar value. Theelement-wise operators are multiplication (.*), division (./), left division (.\) and power (.^).Notice that these are similar to the scalar operators but prefixed with a period.

>> A.*Bans =

0.3820 0.0936 0.8491 0.36780.1163 0.1826 0.1472 0.59470.7236 0.0089 0.4751 0.0557

>> A./Bans =

2.1476 0.1017 1.1777 0.64060.1387 0.4247 0.1688 1.07691.1530 7.0005 1.0313 0.3617

>> A.^Bans =

0.9591 0.1072 1.0000 0.57830.1511 0.4325 0.1781 0.84740.9307 0.9517 0.7850 0.4649

Code Example 28: Element-wise arithmetic


3.3.4 Transpose

MATLAB has two matrix suffix operators (and matching built-in functions) to take the transposeof a matrix. The complex conjugate transpose operator (single-quote ’ or the ctranspose function)takes the transpose of a matrix and the complex conjugate in one operation; whereas, the transposeoperator (.’ or the transpose function) just takes the transpose. For real-valued matrices thesetwo operators are equivalent.

>> A’ans =

0.9058 0.1270 0.91340.0975 0.2785 0.25001.0000 0.1576 0.70000.4854 0.8003 0.1419

>> A.’ans =

0.9058 0.1270 0.91340.0975 0.2785 0.25001.0000 0.1576 0.70000.4854 0.8003 0.1419

>> Z = complex(A,B)Z =

0.9058 + 0.4218i 0.0975 + 0.9595i 1.0000 + 0.8491i 0.4854 + 0.7577i0.1270 + 0.9157i 0.2785 + 0.6557i 0.1576 + 0.9340i 0.8003 + 0.7431i0.9134 + 0.7922i 0.2500 + 0.0357i 0.7000 + 0.6787i 0.1419 + 0.3922i

>> Z’ans =

0.9058 - 0.4218i 0.1270 - 0.9157i 0.9134 - 0.7922i0.0975 - 0.9595i 0.2785 - 0.6557i 0.2500 - 0.0357i1.0000 - 0.8491i 0.1576 - 0.9340i 0.7000 - 0.6787i0.4854 - 0.7577i 0.8003 - 0.7431i 0.1419 - 0.3922i

>> Z.’ans =

0.9058 + 0.4218i 0.1270 + 0.9157i 0.9134 + 0.7922i0.0975 + 0.9595i 0.2785 + 0.6557i 0.2500 + 0.0357i1.0000 + 0.8491i 0.1576 + 0.9340i 0.7000 + 0.6787i0.4854 + 0.7577i 0.8003 + 0.7431i 0.1419 + 0.3922i

Code Example 29: Matrix transposition

3.3.5 Solving Linear Systems

MATLAB has built-in support for solving linear systems by use of the left division (\) and divi-sion (/) operators. Given two matrices A,B and a vector of unknowns x; then, x = A\B gives thesolution to the equation Ax = B and x = B/A gives the solution to the equation xA = B. Notethat in the first case A and B require the same number of rows and in the second case A and Brequire the same number of columns.

>> A = [3 1 -1; 1 1 1; 0 1 -1]A =

3 1 -11 1 10 1 -1

>> B = [0;0;1]B =

001


>> A\Bans =

-0.33330.6667-0.3333

>> B’/Aans =

-0.1667 0.5000 -0.3333

Code Example 30: Solving linear systems

3.4 Functions

All basic mathematical functions in Section 2.5 can be applied (usually element-wise) to matricesand vectors.

3.4.1 Vector Functions

Below is a non-exhaustive list of functions for vectors. These can also be applied to matrices, inwhich case each column of the matrix is treated as a different vector by default, returning a rowvector of the results.

min, max Minimum/maximum value in the vectorsum Sum of all valuesprod Product of all values

mean, median Mean/median of the valuesstd, var Standard deviation/variance of the valuescumsum Cumulative sum of the valuescumprod Cumulative product of the valuessort Sorts the values in the vector

3.4.2 Matrix Functions

Below is a non-exhaustive list of functions for matrices. All basic mathematical functions inSection 2.5 can also be applied (usually element-wise) to the matrix.

inv Inverse a matrix (do not use for solving linear systems, see Section 3.3.5)det Calculate the determinant of a matrixtrace Calculate the trace of a matrixnorm Calculate a norm of the matrix (defaults to 2-norm)rank Calculate the rank of the matrixeig Calculate eigenvalues and eigenvectors of the matrixpoly Calculate characteristic polynomial of the matrixcond Calculate the condition number of the matrix

expm, logm Matrix exponential and logarithmsqrtm Square root of the matrix

Some functions in MATLAB return more than one value. One such example is eig. By defaulta multiple function will return a single result (sometimes the first result only). In the case of eigthis will be a column vector containing the eigenvalues of the matrix. In order to obtain all theresults from the function you need to assign the result directly to multiple variables. This is doneby listing the variable names separated by commas and surrounded by square brackets [] on theleft-hand side of the assignment. In the case of eig the two return value version returns a matrixcontaining each eigenvector as a column and a diagonal matrix of the eigenvalues. If you wantto ignore a return value you can use the tilde ~ instead of a function name for that return value.


>> eig(A)ans =

3.36151.1674-1.5289

>> [V,D] = eig(A)V =

-0.9011 0.2579 0.2860-0.4226 -0.8773 -0.4480-0.0969 -0.4048 0.8471

D =3.3615 0 0

0 1.1674 00 0 -1.5289

>> [V,~] = eig(A)V =

-0.9011 0.2579 0.2860-0.4226 -0.8773 -0.4480-0.0969 -0.4048 0.8471

Code Example 31: Handling multiple function returns (calculating eigenvalues/eigenvectors)

4 Strings

MATLAB supports string/character values. A string is essentially a special vector of characters,which can be entered using single quotation marks. As a vector you can access sub-strings andindividual characters by using standard vector indexing.

>> str = ’This is a test string’str =This is a test string

>> str(6)ans =i

>> str(11:14)ans =test

Code Example 32: String example

You can concatenate strings together by surrounding multiple string values/literals with squarebrackets [].

>> newstr = [’Concatenate "’ str ’" in the middle of this string’]newstr =Concatenate "This is a test string" in the middle of this string

Code Example 33: String concatenation

4.1 Formatting Numbers

It is possible to convert numbers into strings using the built-in functions num2str and sprintf.The first function takes a scalar, vector or matrix and converts it into a string (or array of strings)containing the numbers formatted with at most 4 decimal places. An optional second argumentcan take a scalar number to specify the number of decimal places to use, or a format argument(see sprintf). The sprintf function is more complicated (and is based on the function with the


same name from the C programming language). This function takes as the first argument a stringwhich can contain a format specifier (a ’%’ followed by some parameters; for example, ’%08d’formats an integer padded to 8 characters with leading zeros). For each format specifier a valueis read from the rest of the specified arguments (with matrices and vectors being expanded to alist of arguments). The format is applied as many times as necessary to handle all the argumentsspecified. The MATLAB documentation for this function should be read as it contains far moredetail than can be covered here.

>> num2str(2.53380112)ans =2.5338

>> num2str(2.53380112,7)ans =2.533801

>> sprintf(’%08d’,4)ans =00000004

>> sprintf(’Test %d %d %d; ’, [3 1 -1; 1 1 1; 0 1 -1])ans =Test 3 1 0; Test 1 1 1; Test -1 1 -1;

Code Example 34: Formatting numbers

Notice that sprintf evaluates a matrix column-wise (the first three values printed are the valuesfrom the first column).

4.2 Displaying Text

MATLAB by default has outputted the results of its computation in its own format. Using num-ber formatting and strings it is possible to generate customised text output using the disp com-mand, which takes a string and outputs it to the Command Window.

>> x = 1:10;>> disp([’First 10 factorials:’ sprintf(’\n%4d: %8d’, [x; cumprod(x)])])First 10 factorials:

1: 12: 23: 64: 245: 1206: 7207: 50408: 403209: 362880

10: 3628800

Code Example 35: Displaying text

Notice in the sprintf function a ’\n’ in the string inserts a new line in the output. A fprintf

function also exists, which works in a similar manner to sprintf, but rather than return a stringit prints the result directly to the Command Window without a new line at the end.

5 Graphics

MATLAB has support for plotting data in various formats. In this section we shall discuss thebasic plotting functions available.


5.1 Plot Basics

The main basic plotting function is simply called plot. This function is used to plot x and y dataagainst each other as a line plot. The function can take a variable number of arguments, witheach set of three arguments (a triple) detailing a set of data to plot, and how to plot it. The firstargument is the x data for plot, the second argument is the y data and the third argument isa string line specification specifying the format for the line. A line specification consists of threeparts: a colour, a marker and a line type.

Colour Marker Lineb Blue . Point - Solidg Green o Circle : Dottedr Red x Cross -. Dash-dotc Cyan + Plus -- Dashedm Magenta * Star (none) No liney Yellow s Squarek Black d Diamondw White v Triangle (down)

^ Triangle (up)< Triangle (left)> Triangle (right)p Pentagramh Hexagram

After each triple of data we can also specify key-value pairs, where the key is a string, for speci-fying more detailed plot information (use doc plot for more information). The plot function canalso take a single vector argument, in which each item in the vector is plotted against the vectorindex, or just two vector arguments (the x and y values). In these case a default line specificationis used.

>> x = 0:0.5:3x =

0 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000

>> plot(x,x,’-xr’,x,x.^2,’-ob’,x,x.^3,’-sk’)

Code Example 36: Basic plotting

0 0.5 1 1.5 2 2.5 3

0

5

10

15

20

25

30

Figure 2: Basic plotting result

Here, MATLAB has automatically decided the minimum and maximum values to display on thex and y axis. Three functions exist for changing these limits. xlim([min max]) sets the limits of


the x axis, ylim([min max]) sets the limits of the y axis, and axis([xmin xmax ymin ymax]) sets thelimits of both axis. All these functions are applied to the currently active plot.

plot uses a linear x and y axis. There also exists the functions loglog, semilogx, and semilogy,which use logarithmic xy, logarithmic x/linear y, and linear x/logarithmic y, respectively.

5.2 Annotation

In the previous example of plotting you will notice that there is no title or axis labels. We canadd these by use of the title, xlabel and ylabel commands. Note that basic TEX support isimplemented in the string specified in the arguments (full LATEX support can also be enabled).We would also like to be able to add a legend to the plot. We do this with a the legend functionthat takes a list of strings as the legend for each plot (the first string is the first plot, etc.). Thelegend can also take key-value pairs for more detailed settings. The most important of these isthe ’Location’ value, which allows us to position the legend on the plot.

>> xlabel(’x’)>> ylabel(’f(x)’)>> title(’Basic Polynomial Functions’)>> legend(’x’,’x^2’,’x^3’,’Location’,’NorthWest’)

Code Example 37: Annotating a plot

x

0 0.5 1 1.5 2 2.5 3

f(x)

0

5

10

15

20

25

30Basic Polynomial Functions

x

x2

x3

Figure 3: Annotated plotting result

As can be seen here, the font can be a little small, this can be changed with key-value pairarguments to all the annotation functions. However, often a simpler way is to use the interac-tive Plot Tools. To open this on an active plot, either enter plottools at the prompt, or use theShow Plot Tools and Dock Figure tool-bar button ( ) on the plot window. You can see the MATLABcode required to generate a plot by clicking the File Generate Code... menu item.

5.3 3D Plots

MATLAB also has a number of functions for plotting three-dimensional data (x, y and z data).The first we shall consider is the plot3 function, used to plot a 3D line plot. This function worksin a similar manner to plot, except you need to provide vectors of x, y and z data for each line.

>> t = linspace(0,10*pi,501);>> plot3(sin(t),cos(t),t,’-r’)

Code Example 38: 3D line (parametric) plot


1

0.5

0

-0.5

-1-1

-0.5

0

0.5

25

0

5

10

20

15

35

30

1

Figure 4: 3D line (parametric) plot

The annotations work as before, although we also have a zlabel function for labelling the z axis.We can also plot full 3D functions using the surf plot. The normal usage of this function takes

three arguments, for the x, y and z data. The z data must be a M × N matrix; whereas, x and ycan be a matrix or a vector (but both must be the same). If x and y are matrices they must be thesame size as the z matrix and each matching element from the three vectors denotes a x, y and zpoint to plot on the surface. If x and y are vectors then x must be of length N and y must be oflength M ; in this case, each point plotted is (x(j), y(i), z(i, j). MATLAB has an in-built functioncalled meshgrid, which takes a x and y vector and creates a full x and y matrix representing thetensor points.

>> x = linspace(0,1,5);>> [X, Y] = meshgrid(x,x)X =

0 0.2500 0.5000 0.7500 1.00000 0.2500 0.5000 0.7500 1.00000 0.2500 0.5000 0.7500 1.00000 0.2500 0.5000 0.7500 1.00000 0.2500 0.5000 0.7500 1.0000

Y =0 0 0 0 0

0.2500 0.2500 0.2500 0.2500 0.25000.5000 0.5000 0.5000 0.5000 0.50000.7500 0.7500 0.7500 0.7500 0.75001.0000 1.0000 1.0000 1.0000 1.0000

>> x = linspace(0,1,25);>> [X, Y] = meshgrid(x,x);>> surf(X, Y, X.*Y.*(1-X).*(1-Y))>> colorbar

Code Example 39: 3D surface plot

We have used the colorbar function to display a colour bar on the active plot. We can changethe colour scheme used by the colormap function, which either takes a string specifying a built-incolour map or a N × 3 matrix of colours (each row is a Red-Green-Blue colour triplet, each valuebetween 0 and 1). See help graph3d for a list available colour maps.

The Rotate 3D tool-bar button ( ) in the plot window allows you to rotate the plot by draggingwithin the plot area. The view(2) or view(3) function calls will switch the current plot view to a2D top-down view or the default 3D view, respectively.

The x and y arguments to surf can be omitted, in which case the values in z are plotted againsttheir indices. The colour of the surface plot is by default calculated from the z value. Optionally, a


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Figure 5: 3D surface plot

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Figure 6: Result of view(2) on 3D surface plot

fourth colour argument can be specified, which specifies the colour (the colour argument of eachpoint is a single value and MATLAB interpolates the colour in the same way as when using zdata).

MATLAB contains several more useful functions for plotting 3D data. The mesh function issimilar to surf, except it only plots the mesh lines (this time coloured), rather than filled polygons.The trisurf and trimesh functions take only vector x, y, and z data, but also takes as the firstargument a M × 3 matrix of triangles; this matrix denotes in each row the indices (in the x, y andz data) of a single triangle to plot. The contour function is similar to surf, but plots a 2D contourplot; an optional fourth argument can be used to specify the number of contour levels. Finally,surfc and meshc draws a surface or mesh plot, respectively, with a contour plot underneath.

>> x = linspace(-pi,pi,25);>> [X, Y] = meshgrid(x,x);>> meshc(X, Y, sin(X).*sin(Y))

Code Example 40: 3D mesh/contour plot


3

2

1

0

-1

-2

-3-3

-2

-1

0

1

2

3

1

-1

-0.5

0

0.5

Figure 7: 3D mesh/contour plot

6 Programming

So far we have only run MATLAB commands one at a time within the Command Window. Whilethis is useful for testing and learning proposes we need to be able to run multiple commands asa single program.

6.1 Scripts

The simplest method for running multiple commands is the use of MATLAB scripts. A MATLABscript is basically a simple text file (with a .m extension) filled with MATLAB commands to exe-cute. Scripts are executed by typing the name of the file (without extension) at the prompt. Thescript is found by looking in the path and current folder (as described in Section 1.2). Note thatthe current folder is searched first, so a file in the local folder with the same name as a built-inMATLAB function will overwrite the MATLAB function (so a script file called sin will result inall calls to the sine function producing incorrect results). All variables and values in the script fileare saved into the workspace, so the values can be accessed after execution, and the script canaccess any variables currently in the workspace.

In order to start editing a script file within MATLAB you can either create a new file withHome New Script or Home New Script menu bar buttons, or you can type edit filename intothe Command Window (where filename is the name of the file with or without extension). In thefirst case the file will not be created until you save it. In the later case, if a script with the specifiedname already exists it will be opened for editing; otherwise, a prompt will ask if you want tocreate a script with that name. In either case the Editor window will appear with a blank text file,simple enter the script and save.

MATLAB scripts may contain comments (lines of text which are not executed) and blanklines. Comments in MATLAB start with a percentage sign % and continue until the end of theline. When run the script will output the result of any command not ending in a semi-colon ;

(the same as if run from the Command Window); therefore, it is advisable in scripts to always endlines with a semi-colon. With scripts it is desirable to keep lines short (so they can be easily read);in fact, the MATLAB Editor window includes a line down the page to indicate a sensible linelength (comments are automatically split at this point). If you enter a long command and needto split it over multiple lines you need to use the line continuation ellipsis ... (three periods) at theend of the line to continue. Note that you cannot use this in the middle a string (split the stringand use string concatenation, see Section 4).

Below is a sample script to demonstrate a few of these point, note the comments and thefprintf function call which is split over two lines. Create a script file sample_script.m in a local


directory (you may want a separate directory for this course) containing the following code.

1 % This is a comment23 % Set up variables4 A = [3 1 -1; 1 1 1; 0 1 -1];5 B = [0;0;1];67 % Print out some matrix properties8 disp(’Properties of A:’);9 disp([’ Condition number: ’ num2str(cond(A))]);10 disp([’ Determinant: ’ num2str(det(A))]);11 % Following command is split over two lines12 fprintf(’ Characteristic Polynomial: %.1fx^3%+.1fx^2%+.1fx%+.1f\n\n’, ...13 charpoly(A));1415 % Solve Ax=b16 disp(’Solving Ax=b’);17 x = A\B1819 % Display the output20 disp([’x = [’ sprintf(’%8.4f’, x) ’]’]);

Code Example 41: Sample script

To execute the script, do one of the two following options:

• Ensure you are in the directory containing the file (cd as necessary) and enter sample_scriptinto the Command Window, or

• Click the Run button in the Editor tab of the Editor window with the script file open (if youare not in the correct folder MATLAB will present a warning dialog — select Change Folderfrom this dialog).

Either method will run the code and output the results.

>> sample_scriptProperties of A:

Condition number: 3.2355Determinant: -6Characteristic Polynomial: 1.0x^3-3.0x^2-3.0x+6.0

Solving Ax=bx = [ -0.3333 0.6667 -0.3333]

Code Example 42: Sample script output

The Editor underlines warnings (in orange) and errors (in red) within a script file, with a match-ing symbol in the right margin. Hovering over these with the cursor will display a message onthe problem, and a button to potentially fix the problem (if MATLAB can calculate an obviousfix). Errors are problems that will stop the script running (for example forgetting the ... whensplitting over two lines the command in the above example). Warnings are things that MATLABconsiders bad practice (forgetting a semi-colon at the end of an assignment), but will still allowthe code to be run.

6.2 Functions

So far we have only used built-in MATLAB functions. MATLAB as a programming languageallows us to define our own functions as necessary. These are defined in a script file similar tobasic scripts, but with a specific format (a required header line defining the function). You cancreate a function by editing a script file as specified above and entering the necessary code, or tohave MATLAB automatically generate a template use the Home New Function menu bar button.


1 function [ output_args ] = functionname( input_args )2 %FUNCTIONNAME Summary of this function goes here3 % Detailed explanation goes here45 end

Code Example 43: Default function structure

The above shows an example of an automatically generated function template. This will needediting as required. The definition of the function breaks down into the following components:

function Keyword Keyword to denote we are writing a function.

output_args Comma-separated list of output argument names. If the function has no out-put then the code between the function keyword and the functionname can be omitted([ output_args ] =).

functionname The name of the function (used when calling the function). The file name of thefile containing the function must be functionname.m (MATLAB will display a warning if thefile name and function name differ).

input_args Comma-separated list of input argument names.

H1 Comment Line The first comment line immediately following the function definition line.Should contain the function name followed by a very short description.

Further documentation comments More comment lines immediately following the function def-inition. The text in these comment lines, along with the H1 Comment Line, will be displayedwhen doc functionname or help functionname are called.

Main Body All code between the function definition and the matching end keyword. Makes upthe code executed when the function is called

end Keyword The end of the function definition. Note this is optional if this is the only functiondefined within the file.

Functions variables have scope only to the function. This means that the function does not haveaccess to any variable in the workspace, and any variables set in the function are not availablein the workspace. To allow arguments to be passed to and from your function you specify alist of input arguments in the brackets after the function name and a list of output argumentsin the square brackets after the function keyword. The input arguments will have the values ofwhatever is passed when the function is called. To return values in the output arguments, simplyassign the result to return to the variable name of the output argument. When the function iscalled the caller specifies how many return values to handle (see Section 3.4.2) by the variablesthat the result is assigned to. If you function returns more variables then these are ignored. Aspecial variable nargout is available to functions, which gives the number of return values thecaller requested. You can also allow a variable number of input arguments, but that is beyondthe scope of this course.

1 function [ z, w ] = sample_function( x, y )2 %SAMPLE_FUNCTION Calculates the product and difference of two numbers3 z = x*y;4 w = x-y;5 end

Code Example 44: Simple function

The above sample show the definition of a function with two returns. This function is called likeany other MATLAB function (providing it is in the current folder/path).


>> x = sample_function(2,3)x =

6

>> [x,y] = sample_function(2,3)x =

6

y =-1

Code Example 45: Calling the simple function

There is one problem with the function. While it works for scalar inputs it will fail with vector ormatrix based inputs. Often when we write functions we want them to be as generic as possible(notice that sin, for example, is defined for scalar, matrix or vector inputs). We can fix this byusing the element-wise operators in preference to the normal scalar operators.

1 function [ z, w ] = sample_function_vec( x, y )2 %SAMPLE_FUNCTION_VEC Calculates the product and difference of two inputs3 z = x.*y;4 w = x-y;5 end

Code Example 46: Simple function changed for vector/matrix inputs

>> [x,y] = sample_function([1 5],[2 5])Error using *Inner matrix dimensions must agree.

Error in sample_function (line 3)z = x*y;

>> [x,y] = sample_function_vec([1 5],[2 5])x =

2 25

y =-1 0

Code Example 47: Calling functions with vector arguments

Function files can actually contain more than one function; however, only the first “main”function defined in a file is visible outside the file. This allows you to write local functions usedby the main function in the file but not usable by anyone else.

1 function [ z, w ] = sample_subfunction( x, y )2 %SAMPLE_SUBFUNCTION Calculates the product and difference of two inputs3 z = multiply(x, y);4 w = x-y;5 end67 function [z] = multiply(x,y)8 %MULTIPLY Elementwise multiplication of the vectors9 z = x.*y;10 end

Code Example 48: Sub-function example

In this case the multiply is only available to the sample_subfunction function. Note that we canstill have comments after the function definition of a sub-function. To see the documentation youneed to specify both the main function name and the sub-function name separated by a > in thecall to help or doc. For example, to see the documentation of the multiply sub-function simplycall help sample_subfunction>multiply.


A special statement, return, can be used anywhere in a function. When this statement isexecuted MATLAB exits the current function and executes no more commands from the function.This can be combined with if statements (see Section 6.5) to handle special cases or abort early.Unlike in some other programming languages return does not take any arguments (a return value), asreturn values are set by assignment.

6.3 Logical Operators

MATLAB has a built-in logical type (often called boolean type in other language), which can takea true or false value only (represented in MATLAB by 1 and 0, respectively). This logical typeis returned when using any of the MATLAB comparison operators. These comparison operatorsare performed element-wise on matrices of the same size and return a matrix with 1 (true) or 0(false) in each entry indicating if the condition is true for the matching elements. The comparisonoperators can be used with one scalar and one vector/matrix argument, in which case the scalaris expanded into a matrix of the correct size.

A < B Checks if A < B (element-wise)A > B Checks if A > B (element-wise)A <= B Checks if A ≤ B (element-wise)A >= B Checks if A ≥ B (element-wise)A == B Checks if A = B (element-wise)A ~= B Checks if A 6= B (element-wise)

The first four of these work only on the real part of a complex number; whereas, == and ~= workson both the real and imaginary parts. In Section 2.2 we mentioned that double-precision floatingpoints suffer from rounding error. The main issue this results in is that the equality comparisonsdo not always return true when we expect, as although two numbers may appear to be the samethere may be a small difference. For example, on my machine, the following occurs:

>> (19.2-19) == 0.2ans =

0

>> 19.2-19-0.2ans =

-7.216449660063518e-16

Code Example 49: Example of rounding error in comparisons

Generally, if we are dealing with a comparison of floating point numbers we calculate the abso-lute value of the difference between the numbers and check if it less than some tolerance value(a small number). For example, rather than evaluate A==B, we instead use, abs(A-B) < 1e-8. Italso worth noting that the special value NaN always compares as not equal to any value, includingitself (NaN == NaN returns false). To detect NaN values you should use the isnan function.

The comparison operators above work on numbers. MATLAB has two functions, strcmp andstrcmpi, which perform case sensitive and case insensitive, respectively, comparisons (equality) ontwo strings.

MATLAB has two functions, called true and false, which work in a similar way to ones andzeros, respectively, except that the return values are logical rather than double. These functionscan also be used without arguments to return a scalar true or false value.

MATLAB also has a number of logical operators and functions, which act (element-wise) onmatrices of logical values.

Operator Function Description~A not(A) Performs a logical NOT (0 becomes 1, 1 becomes 0)

A & B and(A,B) Performs logical AND (returns 1 if both A and B are 1)A | B or(A,B) Performs logical OR (returns 1 if either A or B are 1)

xor(A,B) Performs logical XOR (returns 1 if only one of A and B are 1)


Two functions (all and any) exist, which can be applied to a matrix or vector. Applied to a vectorthese functions perform a logical AND or OR, respectively, to the elements of the vector; whenapplied to a matrix the logical AND or OR is applied to each column (returning a row vector).

>> x = rand(3)x =

0.2599 0.1818 0.86930.8001 0.2638 0.57970.4314 0.1455 0.5499

>> y = x > 0.5y =

0 0 11 0 10 0 1

>> z = x < 0.2z =

0 1 00 0 00 1 0

>> y | zans =

0 1 11 0 10 1 1

>> w = any(y)w =

1 0 1

>> all(w)ans =

0

Code Example 50: Logical operations

Two, short-circuit logical operators also exist. The short-circuit logical AND && and short-circuitlogical OR || can only be applied to scalar arguments. This functions are called short-circuit be-cause they only evaluate the second argument if necessary. Basically, for the short-circuit logicalAND the second argument will not be evaluated if the first argument is false and for the short-circuit logical OR the second argument will not be evaluated if the first argument is true (becausein both cases MATLAB already knows the result of the logical operator).

The main use of logical values will arise in the next few sections; however, they can also beused to index into a vector/matrix. If a logical matrix is passed to the index argument of a matrixof the same size it will return only the values whose matching logical element is true.

>> x = rand(4)x =

0.7803 0.0965 0.5752 0.82120.3897 0.1320 0.0598 0.01540.2417 0.9421 0.2348 0.04300.4039 0.9561 0.3532 0.1690

>> x(x < 0.5) = 0x =

0.7803 0 0.5752 0.82120 0 0 00 0.9421 0 00 0.9561 0 0

Code Example 51: Using logical indexing


A find function also exists in MATLAB which returns the indices of all non-zero entries in amatrix or vector. This can be passed a logical matrix to find all entries in a matrix which meet acertain condition.

>> [i,j] = find(x < 0.02)i =

2

j =4

Code Example 52: find indices of all values < 0.02 in matrix

6.4 Loops

So far we have only considered statements which are executed once. Loops are a programmingstructure that allows us to execute a number of statements several times. MATLAB has bothfor and while loop types. An important note is that although we can use loops to iterate through theelements in a vector/matrix and handle them one at a time this should be avoided wherever possible (byusing element-wise and matrix operations) for performance (speed) reasons.

6.4.1 for Statement

A for loop executes a set of statements a set number of times, each time with an index variablewhich takes the next value from an vector/matrix. The general structure of the for loop is:

1 for index = values2 statements3 end

Code Example 53: for loop structure

Here, index is a variable name to use as the index variable and values is the vector/matrix ofvalues to take. The first time the statements are executed index will be a column vector contain-ing the first column of values, the second time statements are executed index will be a columnvector containing the second column of values, etc.. In normal usage values is a row vector, soeach iteration index takes a single value; in fact, usually we use the start:end or start:step:end

notation directly.

1 disp(’The first 10 factorials’);2 v = 1;3 for i = 1:104 v = v*i;5 fprintf(’%3d : %8d\n’, i, v);6 end

Code Example 54: Factorial calculation with for loop

The first 10 factorials1 : 12 : 23 : 64 : 245 : 1206 : 7207 : 50408 : 403209 : 36288010 : 3628800

Code Example 55: Result of factorial calculation with for loop


Notice that the statements within the for loop are indented. This is sensible to do with any block-style statement as it makes the code easier to read (the MATLAB editor will actually automaticallyindent for you, and unindent when you enter end).

Within a for loop, two special statements can be used. break exits the loop completely (nomore values from values list are evaluated) when it is called and continue stops executing thestatements for the current value and moves to the next value from values (if one exists).

6.4.2 while Statement

The while loop continues to execute the statements it contains while some condition holds true.

1 while expression2 statements3 end

Code Example 56: while loop structure

When this statement is reached the expression, which should return a logical matrix, is evaluated.If all values in the matrix are non-zero (true) then the statements are executed. The expression

is then evaluated again and statements executed if all entries of the matrix are non-zero. Thiscontinues until one entry of expression evaluates to 0 (false).

1 v = 100;2 while v > 0.53 disp(num2str(v,8));4 v = v/2;5 end

Code Example 57: while loop example

100502512.56.253.1251.56250.78125

Code Example 58: Result of while loop example

If the expression never evaluates to false then the code will be stuck in an infinite loop and thescript will never terminate. You can force a running script to terminate by using the Ctrl + C

keyboard shortcut within the Command Window.Within a while loop break and continue can be used in similar manner to a for loop. break

exits the loop, continue stops executing the statements and evaluates expression again.

6.5 if-else Statement

Using the logical values we have seen in Section 6.3 is it possible to write MATLAB code whichonly executes statements if a particular condition is true. To do this we use a if statement, whichhas the general form:

1 if expression2 if_statements3 elseif expression4 elseif_statements5 else6 else_statements7 end

Code Example 59: if-elseif-else structure


The elseif block (with its following statements) and the else block (with its following state-ments) are both optional. You can also have multiple elseif statements (must be after the if state-ment and before the else statement). When an if statement is reached the (logical) expressionis evaluated. If all values in the resulting matrix are non-zero (true) then if_statements are ex-ecuted; however, if expression evaluates to false then the expression for the first elseif state-ment is evaluated and elseif_statements executed if this is true. This continues, evaluating theexpression for each elseif statement, in order, until one is true (then the matching statements areexecuted). If no expression evaluates to true then the else_statements are executed.

1 function plotdata(x, y, xtype, ytype)2 % plotdata Plots the data on linear or log plots3 % xtype and ytype are strings specify the type of4 % scale for that axis - either ’log’ or ’linear’.5 if (strcmpi(xtype,’log’) && strcmpi(ytype,’log’))6 loglog(x, y);7 elseif strcmpi(xtype,’log’)8 semilogx(x, y);9 elseif strcmpi(ytype,’log’)10 semilogy(x, y);11 else12 plot(x, y);13 end14 end

Code Example 60: if example

6.6 switch Statement

The switch statement selects a set of statements to execute based on the value of a number orstring. The structure of the switch statement is as follows.

1 switch expression2 case case_expression3 statements4 otherwise5 otherwise_statements6 end

Code Example 61: switch structure

The otherwise statement is optional and you can have multiple case statements. When reached aswitch statement evaluates the expression and compares the result against all case_expressions.The statements of the first matching case_expression are executed. If no match occurs then theotherwise_statements are evaluated. A case_expression can be a single value, or multiple values(comma-separated and surrounded by braces {}) if multiple values should have the same state-ments executed. Note that unlike in some other programming languages (C/C++ etc.) case statementsdo not have fall-through behaviour and MUST NOT contain a break statement.

1 function [city] = capital(country)2 switch country3 case ’Austria’4 city = ’Vienna’;5 case ’Germany’6 city = ’Berlin’;7 case {’United Kingdom’,’Great Britain’}8 city = ’London’;9 otherwise10 city = ’<Unknown>’;11 end12 end

Code Example 62: switch example

Note that the string comparison here is case sensitive.


6.7 Function Handles & Anonymous Functions

So far all variables/values we have considered have been some form of number or a string. It ispossible, however, to store a reference to a function (called function handles in MATLAB terminol-ogy) within a variable. You can then call that function via the variable by just that variable namelike a function. To take a handle of a function just use the function name prefixed by a @ symbol.

>> sinhandle = @sinsinhandle =

@sin

>> sinhandle(pi/2)ans =

1

Code Example 63: Taking and using function handle of sin

The main use for this is that it allows generic functions to be written that operate on a function,without having to know what function it operates on. If you write such a function you just treatthe specific argument as a function handle and you only need to know how many argumentsto pass to the function. (Note that the function comments should document this for other usersof your functions). MATLAB has several functions that take function handles, such as the easyplotter functions. Almost all plotting functions we discussed in Section 5 have a easy versionwith the same name prefixed by ez, such as ezsurf, ezplot, ezcontour, etc.. This functions takea function handle (or a string containing MATLAB code to evaluate) and plots the function overthe interval [−2π, 2π] (in each coordinate direction), picking the points it samples the function atitself. You can also specify a vector as a second argument to change the range. The number ofarguments the function handle takes depends on the plotting routine (ezplot takes one argumentand ezsurf takes two arguments).

>> ezplot(@sin,[-pi pi]);

Code Example 64: Calling ezplot with function handle of sin

We can use function handles to functions we have written as well. For example we can callezsurf on the sample_function and sample_function_vec functions we wrote in Section 6.2.

>> ezsurf(@sample_function)Warning: Function failed to evaluate on array inputs; vectorizing thefunction may speed up its evaluation and avoid the need to loop over arrayelements.> In ezplotfeval (line 56)In ezgraph3>ezeval (line 635)

...>> ezsurf(@sample_function_vec)

Code Example 65: Calling ezsurf with handle to own function

In Section 6.2 we mentioned that functions should be written as generic as possible, we have hereanother demonstration of why. The ezsurf function has tried to call the function with vector ormatrix inputs (essentially one call with all points it wants to evaluate), but as that failed in thefirst call it then fell back to one point at a time.

Using function handles we can also define anonymous functions. This are functions that arewritten inline in MATLAB (usually fairly simple one-line functions). An anonymous function iswritten as @(input_args) functioncode, where input_args is the same as for a normal functionand functioncode is a line of MATLAB code which returns a result (which is the result of thefunction). You can use this anonymous function like a normal function handle (passing to afunction or assigning to a variable). We can, for example, write Code Example 40 as follows.

>> ezmeshc(@(x, y) sin(x).*sin(y), [-pi pi])

Code Example 66: 3D mesh/contour plot using anonymous functions


7 Structures

MATLAB has support for various types of more complex data structures, which are beyond thescope of this course. One basic complex data type is the struct type. A structure is essentially agroup of variables stored together in a single object. The various variables (fields) in a structurecan be different types. A structure type can be generated in two different ways, either with thestruct function, or by direct assignment of fields. The struct function takes a variable numberof values, where each pair is a key-value pair (the first value is the key and MUST be a string, thesecond is the value).

>> course = struct(’Name’,’Numerische Mathematik 1’,...’Group’,3,’Year’,2015,’Semester’,’WS’)

course =Name: ’Numerische Mathematik 1’Group: 3Year: 2015

Semester: ’WS’

Code Example 67: Generating structure using struct function

You can access a field, for reading or assignment of value, by using the dot . notation. Here, youuse the variable name, followed by a period and then the name of the field.

>> course.Name = ’Numerische Mathematik 1’;>> course.Group = 3;>> course.Year = 2015;>> course.Semester = ’WS’;>> coursecourse =

Name: ’Numerische Mathematik 1’Group: 3Year: 2015

Semester: ’WS’

>> course.Nameans =Numerische Mathematik 1

Code Example 68: Generating and reading structure directly

You can also create structure arrays. Structure arrays are accessed in the same way as vectors,and assigning fields to an index that doesn’t currently exist will grow the array to the correct size.

>> course(3).Group = 4course =1x3 struct array with fields:

NameGroupYearSemester

>> course(2)ans =

Name: []Group: []Year: []

Semester: []

>> course(3)ans =

Name: []Group: 4Year: []

Semester: []

Code Example 69: Accessing structure array


Structures, and structure arrays, can be nested with a structure field containing another structureor array.

8 Error Handling

In some of the code examples above you will have noticed red and orange messages appearing inthe Command Window. This are errors, and warnings, that occurred while trying to run the code.In this section we shall discuss how to handle errors, and incorrect results.

8.1 Understanding Error Messages

When an error message appears the key is understanding what it is trying to explain. In mostcases this is fairly obvious (although occasionally a different error message to the real problemmay appear).

>> A = rand(4);>> B = rand(5);>> A*BError using *Inner matrix dimensions must agree.

>> sample_functionError using sample_function (line 3)Not enough input arguments.

Code Example 70: Example error messages

In the case of the first error, we are trying to matrix multiple two matrices with incompatibledimensions (a 4× 4 with a 5× 5), so MATLAB complains about Inner matrix dimensions, as theinner matrix dimensions are the number of columns in the first matrix and the number of rows inthe second matrix. In the second case we have tried to call a function without the correct numberof arguments. As this is a function we wrote it has also given us a line number in the functionwhere the error occurred. This line number is a clickable link, which will open the Editor windowat the specified line. (In this case the error is not here, but it is the location were it is detected —the line the arguments are first used).

The second situation highlights a useful feature. Take the following code (see if you can spotthe error before running).

1 function [b] = invalid_func(n)2 % invalid_func Function that we want to take a number3 % and perform Ax for A=rand(n), x=1:n4 A = rand(n);5 x = 1:n;6 b = A*x;7 end

Code Example 71: Function with error

When we try to run this function we get an error.

>> invalid_func(4)Error using *Inner matrix dimensions must agree.

Error in invalid_func (line 6)b = A*x;

Code Example 72: Running function with error


MATLAB has told us the error (matrix multiplication with incorrect dimensions), the line theerror occurred on, and has even printed the line causing the error as well. So if we click the linenumber we can then look at the code and try and find the error. The error is a matrix dimensionproblem so we need to look at the sizes of A and x. A=rand(n), so that is a n× n matrix, which isas we expect. x=1:n, so that is a vector of n values; however, it is a row vector, or a 1 × n matrix.So the solution is just to transpose x.

8.2 Generating Errors

When you are writing a code it is possible that you will want to generate error messages incertain cases. The most common of these is checking that input values to a function are valid.When you want to generate an error call the error function, passing a error message (string) todisplay. When this call is executed the function terminates and the error message is displayed inthe Command Window.

1 function [x] = basic_factorial(n)2 % basic_factorial A very basic (and naive) factorial implementation34 x = 1;5 if (n < 0)6 error(’Factorial only defined for non-negative numbers’);7 elseif (round(n) ~= n)8 error(’Factorial only defined for integer values’);9 elseif (n > 0)10 for i=1:n11 x = x*i;12 end13 end1415 end

Code Example 73: Generating errors

>> basic_factorial(-2)Error using basic_factorial (line 6)Factorial only defined for non-negative numbers

>> basic_factorial(2.3)Error using basic_factorial (line 8)Factorial only defined for integer values

>> basic_factorial(0)ans =

1

>> basic_factorial(4)ans =

24

Code Example 74: Executing a function with error checking

The error function can also take a string, followed by a avriable number of arguments. In thiscase it treats the string and the arguments in the same way as sprintf. A warning function alsoexists that takes identically arguments to error. This function generates a warning message inthe Command Window when executed, but allows the script to continue running.

8.3 Debugging

When running scripts or functions it is possible an error occurs you were not expecting, or anunexpected value is returned. MATLAB has a debugger, which allows you to run the code andinspect the values within a function. The best way to enter debug mode is to place a breakpoint


on a line of code where you want the execution to pause and then execute the code as normal.A breakpoint can be placed (or removed) by clicking in the left margin of the Editor window(between the code and the line number) or by using the Editor Breakpoints menu. In both cases,when a breakpoint is active on a line a small red circle will appear in the left margin. When thecode is executed the script will run until this line is reached, and then pause (with the Editorwindow active at that line). You can inspect the value of variables in the script/function byhovering over them with the mouse (a tooltip with the value appears), or by entering the variablename at the K>> prompt in the Command Window. You can also select some code (either just avariable or a whole expression), right-click and select Evaluate Expression . The result is output tothe Command Window. Note that a breakpoint pauses the code before execution of a line (so anyvariables on that line will not exist yet).

Figure 8: Debug tools

In debug mode the Editor tab on the Editor window contains a set of tools for controlling theexecution of the script. Continue will continue running the script until the next breakpoint occurs.Step will execute the current line and then pause on the next line. Step In does the same, but if theline contains a function call then instead it will pause at the first line inside that function. Step Out

will run the rest of the current function and will pause at the line of code after the function call.Finally, Quit Debugging will terminate the currently executing script and exit the debugger.

Literature & Resources

P. Arbenz. Einfürhung in MATLAB. ETH Zürich, 2008. URL http://people.inf.ethz.ch/arbenz/MatlabKurs/matlabintro.pdf.

S. Attaway. Matlab: A Practical Introduction to Programming and Problem Solving. Butterworth-Heinemann, Boston, third edition, 2013. URL http://www.sciencedirect.com/science/book/9780124058767.

T. A. Davis. MATLAB Primer. CRC Press, Boca Raton, eighth edition, 2010.

F. Grupp and F. Grupp. MATLAB 7 für Ingenieure. Oldenbourg Verlag, München, 2004.

F. Haußer and Y. Luchko. Mathematische Modellierung mit MATLAB. Eine praxisorientierteEinführung. Spektrum Akademischer Verlag, Heidelberg, 2011. URL http://www.springerlink.com/content/h153hn/#section=786206&page=1.

B. R. Hunt, R. L. Lipsman, and J. M. Rosenberg. A Guide to MATLAB for Beginners and ExperiencedUsers. Cambridge University Press, Cambridge, third edition, 2014.

W. Huyer. Einfürhung in MATLAB. Universität Wien, 2012. URL http://www.mat.univie.ac.at/~huyer/matlab.pdf.

MathWorks. MATLAB Central, 2015. URL http://de.mathworks.com/matlabcentral.[Online].

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http://people.inf.ethz.ch/arbenz/MatlabKurs/matlabintro.pdf

http://www.sciencedirect.com/science/book/9780124058767

http://www.sciencedirect.com/science/book/9780124058767

http://www.springerlink.com/content/h153hn/#section=786206&page=1

http://www.springerlink.com/content/h153hn/#section=786206&page=1

http://www.mat.univie.ac.at/~huyer/matlab.pdf

http://www.mat.univie.ac.at/~huyer/matlab.pdf

http://de.mathworks.com/matlabcentral